Analysis of a diffusion epidemic SIR model with saturated treatment in a heterogeneous environment

The spread of infectious diseases is profoundly influenced by spatial heterogeneity and the availability of medical resources. While reaction-diffusion epidemic models have been extensively studied, there has been little research on SIR models that incorporate a spatially heterogeneous standard incidence rate and a saturated treatment function, which is crucial for modeling the limited capacity of health care systems. To address this, we proposed a novel diffusive SIR epidemic model in a heterogeneous environment. Methodologically, we defined the basic reproduction number $R_0$ and employed Lyapunov functions, the theory of monotone dynamical systems, and asymptotic analysis to investigate the dynamics. Our key results showed that if $R_0<1$, the disease-free equilibrium is globally asymptotically stable, implying epidemic extinction. If $R_0>1$, the disease becomes uniformly persistent, and an endemic equilibrium exists. Furthermore, we derived the asymptotic profiles of the endemic equilibrium as the diffusion rates of susceptible or infected populations approached zero. Numerical simulations not only validated our theoretical findings but also demonstrated that increasing medical resources or reducing spatial heterogeneity can effectively lower the infection peak and help control the disease. These results provide theoretical guidance for designing effective public health policies.